Statistics Colloquium: STA 290

Title: "Limiting spectral distribution of normalized sample autocovariance matrices of linear time series when $p,n \to \infty$ such that $p/n \to 0$"

We study the empirical spectral distribution (ESD) of the matrices of the form $$ \mathbf{D}_\tau = \sqrt{\frac{n}{p}}\left(\mathbf{C}_\tau -\mathbb{E}(\mathbf{C}_\tau)\right) $$ as $p,n\to \infty$ and $p/n\to 0$, where $\mathbf{C}_\tau$ is the lag-$\tau$ symmetrized sample autocovariance matrix based on $X_1,\ldots,X_n$, where $\{X_t\}$ is a $p$-dimensional stationary linear time series. We establish that under simultaneous diagonalizability of the coefficients of the linear process representation, the ESD of $\mathbf{D}_\tau$ converges to a unique nonrandom limit. In special cases, the limit is characterized as the limiting spectral distribution of a weighted Wigner matrix.